On the approximation of Weierstrass function via superoscillations

Source abstract

Abstract Superoscillations have attracted considerable interest among mathematicians, physicists, and signal-processing engineers over several decades, since the seminal works of Aharonov et al (1988 Phys. Rev. Lett. 60 1351–1354) and of Berry (1994 Quantum Coherence and Reality; in celebration of the 60th Birthday of Yakir Aharonov 55–65) on band-limited functions that locally oscillate faster than their highest Fourier frequency. A particularly intriguing question, posed by Berry and Morley-Short (2017 J. Phys. A: Math. Theor. 49 065203), asks whether such sequences, can reproduce fractals, nowhere-differentiable functions. This question represents an extreme application of superoscillations, well beyond the previously studied regimes of physical interest involving variations on fine but finite scales, and is directly motivated by current applications in physics such as superresolution microscopy and the analysis of sub-wavelength optical and quantum fields. The present paper gives a rigorous affirmative answer for the classical Weierstrass function, and clarifies the exact sense in which the approximation holds.